Drag in couple stress fluids

Summary A general expression is derived for the drag experienced by an axially symmetric body in the slow steady flow of Stokes' couple stress fluids. The result is compared with that obtained for micropolar fluids and then applied to the particular case of flow past a sphere.

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Résumé On a dérivé une relation générale pour la traînée subie par un corps de symétrie axiale dans un courant continu de fluide de Stokes à contraintes couplée. Cette expression est comparée à celle obtenue pour des fluides micropolaires, puis elle est appliquée au cas particulier du flux subi par une sphère.

     

An inhomogeneity problem in couple stress theory

Using Hilbert theory and Mindlin’s couple stress theory, the problem of two-dimensional circular inhomogeneity (when the inserted material is of different size than the size of the cavity and having different elastic constants) is studied in this paper. Stresses could be bounded at infinity. The formulation is valid also for regions other than the circular ones when the elastic constants of inclusion (inserted material) and matrix (outer material) are the same. The problem of circular inclusion when the matrix is finite has also been tackled. Numerical results are in conformity with the fact that the effect of couple stresses is negligible when the ratio of the smallest dimension of the body to the characteristic length is large.     

Exact solutions for two dimensional flows of couple stress fluids

Exact solutions are derived for the class of two dimensional couple stress flows. This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream. The solutions are obtained by applying the so-called inverse method which makes certain hypothesis a priori on the form of the velocity field and pressure without making any on the boundaries of the domain occupied by the fluid. Exact solutions are obtained for both steady and unsteady cases.     

Development of size-dependent consistent couple stress theory of Timoshenko beams

In this paper, a linear size-dependent Timoshenko beam model based on the consistent couple stress theory is developed to capture the size effects. The extended Hamilton's principle is utilized to obtain the governing differential equations and boundary conditions. The general form of boundary conditions and the concentrated loading are employed to determine the exact static/dynamic solution of the beam. Utilizing this solution for the beam's deformation and rotation, the exact shape functions of the consistent couple stress theory (C-CST) is extracted, which leads to the stiffness and mass matrices of a two-node C-CST finite element beam. Due to the complexity and high computational cost of using the exact solution's shape functions, in addition to the Ritz approximate solution, a two primary variable finite element model of C-CST is proposed, and the corresponding general deformation and rotation fields, shape functions, mass and stiffness matrices are calculated. The C-CST is validated by comparing the prediction of different beam models for a benchmark problem. For the fully and partially clamped cantilever, and free-free beams, the size dependency of the formulations is investigated. The static solutions of the classical and consistent couple stress Timoshenko beam models are compared, and a criterion for selecting the proper model is proposed. For a wide range of material properties, the relation between the beam length and length scale parameter is derived. It is shown that the validity domain of the consistent couple stress Timoshenko model barely depends on the beam's constituent material.     

Exact solutions for two dimensional flows of couple stress fluids

Exact solutions are derived for the class of two dimensional couple stress flows. This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream. The solutions are obtained by applying the so-called inverse method which makes certain hypothesis a priori on the form of the velocity field and pressure without making any on the boundaries of the domain occupied by the fluid. Exact solutions are obtained for both steady and unsteady cases.     

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.     

Hydromagnetic fluctuating flow of a couple stress fluid through a porous medium

The equations of a polar fluid of hydromagnetic fluctuating through a porous medium are cast into matrix form using the state space and Laplace transform techniques the resulting formlation is applied to a variety of problems. The solution to a problem of an electrically conducting polar fluid in the presence of a transverse magnetic field and to a probem for the flow between two parallel fixed plates is obtained. The inversion of the Laplace transforms, is carried out using a numerical approach. Numerical results for the velocity, angular velocity distribution and the induced magnetic field are given and illustrated graphically for each problems.     

An indeterminate equation

It is proved that on the curve $$x_0^2 + x_1^2 = t(x_2^2 - x_3^2 ),t(x_0^2 - x_1^2 ) = x_2^2 + x_3^2 $$ there are no k(t)-rational points; here k is an algebraically closed field.     

Refining the definition of indeterminate forms

In mathematics and science dictionaries and encyclopedias, as well as in very many calculus textbooks, seven combinations of 0, 1 and are traditionally used as the definition of ‘indeterminate forms’. This is pedagogically inferior inasmuch as there exist several other forms that students are likely to encounter that are also indeterminate.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory

In the present paper, a non-classical model for functionally graded annular sector microplates under distributed transverse loading is developed based on the modified couple stress theory and the first-order shear deformation plate theory. The model contains a single material length scale parameter which can capture the size effect. The material properties are graded through the thickness of plates according to a power-law distribution of the volume fraction of the constituents. The equilibrium equations and boundary conditions are simultaneously derived from the principle of minimum total potential energy. The system of equilibrium equations is then solved using the generalized differential quadrature method. The effects of length scale parameter, power-law index and geometrical parameters on the bending response of annular sector plates subjected to distributed transverse loading are investigated.     

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.      

Mixed convection flow of couple stress fluid between parallel vertical plates with Hall and Ion-slip effects

An analysis is presented to investigate the Hall and Ion-slip effects on fully developed electrically conducting couple stress fluid flow between vertical parallel plates in the presence of a temperature dependent heat source. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations and then solved using homotopy analysis method (HAM). The effects of the magnetic parameter, Hall parameter, Ion-slip parameter and couple stress fluid parameter on velocity and temperature are discussed and shown graphically.     

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.     

Two classes of sub-optimal shapes for one dimensional slider bearings with couple stress lubricants

Shape optimization of slider bearings operating with couple stress lubricants is performed here for the first time by using a novel direct optimal control approach, which defines the gradient of the film height as a control. The bearing load is maximized. One dimensional Reynolds and energy equations are used. Several constraints are taken into consideration. They avoid the occurrence of cavitation and ensure the validity of the Reynolds equation. The model is validated against a known analytical solution (the Rayleigh step bearing). Two simple design rules are inferred, which yield two different classes of sub-optimal shapes: the multi-stepped bearings and the multi-sloped bearings, respectively. Multi-stepped bearings consist of several steps and the couple stress parameter may affect the constant value of the film height between steps. Multi-sloped bearings consist of several inclined regions and the couple stress fluid parameter may affect the constant value of the film height between regions. The slider bearings operation under variable load is stable. A sensitivity analysis identified the design parameters which have the highest impact on bearing performance. The optimal slider bearing shapes obtained for Newtonian lubricants do not change when most common couple stress fluids are used. Isothermal models may be used successfully at lower values of the couple stress parameter.     

Modeling of AFM with a piezoelectric layer based on the modified couple stress theory with geometric discontinuities

AFM has been one of the most accurate instruments for measuring intermolecular forces and surface topography in the nano-scale. Micro-cantilever (MC) with piezoelectric layer has been used to improve the AFM performance. The Classic Continuum Mechanics (CCM) which currently used to develop the governing equation leads to noticeable errors. Hence, the accuracy of the governing equations for examining the MC vibrational behavior needs to be improved by using a modified model. In response to this need, the Modified Couple Stress theory (MCS) based on the Timoshenko beam model has been employed in this research. The governing equations have been derived using the Hamilton's principle and solved using the Differential Quadrature (DQ) method. In the modeling, the geometric discontinuities resulting from the presence of a piezoelectric layer enclosed between electrodes and MC surface area variations resulting from the connection of the probe to the MC have been considered. Moreover, the coupling effects of piezoelectric on MC stiffness have been taken into account. The results have revealed that the size parameter not only affects the frequency and amplitude but also improves the accuracy of the results when compared with the CCM theory. Moreover, the effects of geometric parameters on the piezoelectric MC frequency have been examined.     

Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder

The unsteady natural convective couple stress fluid flow over a semi-infinite vertical cylinder is analyzed for the homogeneous first-order chemical reaction effect. The couple stress fluid flow model ...